Discrete subsets of R^2 and the associated distance sets

On the structure of subsets of the discrete cube with small edge boundary

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} Δ\mathcal{G}| \leq C…

Upper Covers of Chains and Antichains in Sets of Indecomposable Subsets

We prove that there are arbitrarily large indecomposable ordered sets T with a 2-chain C such that the smallest indecomposable proper superset U of C in T is T itself. Subsequently, we characterize all such indecomposable ordered sets T and 2-chains C. We also prove the same type of result for 2-antichains.

Distance sets of two subsets of vector spaces over finite fields

We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we improve upon the results by Rainer Dietmann. In the case that one of the subsets is a product set, we obtain further improvement on the estimate.

Subsets and Freezing Sets in the Digital Plane

We continue the study of freezing sets for digital images introduced in [4, 2, 3]. We prove methods for obtaining freezing sets for digital images (X, c_i) for X \subset Z^2 and i \in {1, 2}. We give examples to show how these methods can lead to the determination of minimal freezing sets.

Independent sets in the discrete hypercube

In this expository note we describe a proof due to A. Sapozhenko that the number of independent sets in the discrete $d$-dimensional hypercube $Q_d$ is asymptotically $2 \sqrt{e} 2^{2^{d-1}}$ as $d$ tends to infinity.

Discrete sets with minimal moment of inertia

This paper has been withdrawn by the author due to a crucial error in the proof of Theorem 1.

Planar lattice subsets with minimal vertex boundary

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We characterize uniquely minimal sets of X: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of X: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate …

Gray-coding through nested sets

We consider the following combinatorial question. Let $$ S_0 \subset S_1 \subset S_2 \subset ...\subset S_m $$ be nested sets, where #$(S_i) = i$. A move consists of altering one of the sets $S_i$, $1 \le i \le m-1$, in a manner so that the nested condition still holds and #$(S_i)$ is still $i$. Our goal is to find a sequence of moves that exhausts through all subsets of $S_m$ (other than the initial sets $S_i$) with no repeats. We call this "Gray-coding through nested sets" because of the analogy with Frank Gray's theory of exhausting through integers while altering only one bit at a time. Ou…

An isoperimetric inequality for antipodal subsets of the discrete cube

A family of subsets of $\{1,2,\ldots,n\}$ is said to be {\em antipodal} if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of $\{1,2,\ldots,n\}$. Our inequality implies that for any $k \in \mathbb{N}$, among all such families of size $2^k$, a family consisting of the union of a $(k-1)$-dimensional subcube and its antipode has the smallest possible edge boundary.

具有少数子集和的集合

Sets with Few Subset Sums

本文研究了具有少数子集和的集合的稳定性，给出了在不同范围内的精确刻画和上限分析。

Maximal subsets free of arithmetic progressions in arbitrary sets

We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots, n]$. The study continues the work of Komlós, Sulyok, and Szemerédi.

Primitive sets and an Euler phi function for subsets of {1,2,...,n}

A nonempty subset A of {1,2,...,n} is called primitive if gcd(A)=1. Let f(n) and f_k(n) denote, respectively, the number of primitive subsets and the number of primitive subsets of cardinality k of {1,2,...,n}. Recursion formulas and asymptotic estimates are obtained for both functions.

Subsets of finite groups exhibiting additive regularity

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In particular, we show that any sum set must exhibit higher-order regularity and that an abelian sum set is necessarily a reversible difference set. We next develop several general construction techniques under the hypothesis that the over-riding group contains a normal subgroup of order 2. Finally, by exploiting properties of dihedral groups and Frobenius groups, s…

On sets of vectors of a finite vector space in which every subset of basis size is a basis II

This article contains a proof of the MDS conjecture for $k \leq 2p-2$. That is, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subset of $S$ of size $k$ is a basis, where $q=p^h$, $p$ is prime and $q$ is not and $k \leq 2p-2$, then $|S| \leq q+1$. It also contains a short proof of the same fact for $k\leq p$, for all $q$.

关于无三点共线的无限集合

On infinite sets with no $3$ on a line

本文构造了一个无限点集，其中任何子集都能找到无三点共线的常密度子集，但整个集合无法被分成多个无三点共线的子集，从而解决了Erdős等人的问题。

具有禁止子集的自然数集

Sets of natural numbers with proscribed subsets

本文研究了禁止特定子集的自然数集，提出了关于G_A(n)上界的新的理论框架和具体例子，包括等比数列和乘法平方的排除问题。

Subsets of Products of Finite Sets of Positive Upper Density

In this note we prove that for every sequence $(m_q)_{q}$ of positive integers and for every real $0<δ\leqslant1$ there is a sequence $(n_q)_{q}$ of positive integers such that for every sequence $(H_q)_{q}$ of finite sets such that $|H_q|=n_q$ for every $q\in\mathbb{N}$ and for every $D\subseteq \bigcup_k\prod_{q=0}^{k-1}H_q$ with the property that $$\limsup_k \frac{|D\cap \prod_{q=0}^{k-1} H_q|}{|\prod_{q=0}^{k-1}H_q|}\geqslantδ$$ there is a sequence $(J_q)_{q}$, where $J_q\subseteq H_q$ and $|J_q|=m_q$ for all $q$, such that $\prod_{q=0}^{k-1}J_q\subseteq D$ for infinitely many $k.$ This gi…

Flat-containing and shift-blocking sets in $F_2^r$

For non-negative integers $r\ge d$, how small can a subset $C\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a $d$-flat passing through $v$ and contained in $C\cup\{v\}$? Equivalently, how large can a subset $B\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a linear $d$-subspace not blocked non-trivially by the translate $B+v$? A number of lower and upper bounds are obtained.

具有不同子集乘积的集合的紧界

Sharp Bounds for Sets with Distinct Subset Products

本文研究了具有不同子集乘积的集合的界，通过素数计数函数给出了新的上界结果。

Additive and multiplicative Sidon sets

We give a construction of a set $A \subset \mathbb N$ such that any subset $A' \subset A$ with $|A'| \gg |A|^{2/3}$ is neither an additive nor multiplicative Sidon set. In doing so, we refute a conjecture of Klurman and Pohoata.

On the Statistics of the Number of Fixed-Dimensional Subcubes in a Random Subset of the n-Dimensional Discrete Unit Cube

This paper consists of two independent, but related parts. In the first part we show how to use symbolic computation to derive explicit expressions for the first few moments of the number of implicants that a random Boolean function has, or equivalently the number of fixed-dimensional subcubes contained in a random subset of the $n$-dimensional cube. These explicit expressions suggest, but do not prove, that these random variables are always asymptotically normal. The second part presents a full, human-generated proof, of this asymptotic normality, first proved by Urszula Konieczna.

自然数子集中的随机有限和集与积集

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers

本文研究了自然数Bernoulli随机子集中加法与乘法结构的出现，证明了几乎必然存在任意有限长度的FS-sets和FP-sets，并建立了Hindman分割定理与中心极限定理的新联系。

On distances and metrics in discrete ordered sets

Discrete partially ordered sets can be turned into distance spaces in several ways. The distance functions may or may not satisfy the triangle inequality, and restriction of the distance to finite chains may or may not coincide with the natural, difference-of-height distance measured in a chain. For semilattices, a semimodularity condition ensures the good behavior of the distances considered. This condition is trivially satisfied by trees, and in lattices it coincides with the usual semimodularity property. For a large class of distance functions the triangle inequality is equivalent to semim…

立方体集合的离散同伦群

Discrete homotopy groups of cubical sets

将图的离散同伦群推广到任意立方体集合，证明拟对称立方体集合的离散同伦群与其几何实现的同伦群自然同构，并给出遗忘函子左伴随的纯组合构造。

关于构造一族在取子集运算下封闭的正整数集合

On the construction of a family of sets of positive integers closed under taking subsets

本文构造了字母表{0,1}上的单侧无穷序列与一类在取子集运算下封闭的正整数集合族之间的显式双射及其逆映射，并利用该双射通过离散动力系统刻画了特定集合族的极值性质。

Almost isoperimetric subsets of the discrete cube

We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + ε)$ can be made into a subcube by at most $(2 ε/\log_{2}(1/ε))|A|$ additions and deletions, provided $ε$ is less than an absolute constant. We deduce that if $A \subset \{0,1\}^{n}$ has size $2^{t}$ for some $t \in \mathbb{N}$, and cannot be made into a subcube by fewer than $δ|A|$ additions and deletions, then its edge-boundary has size at least $|A| \log_{2}(2^{n}/|A|) + |A| δ\log_{2}(1/δ) = 2^{t}(n-t+δ\log_{2}(1/δ))$, provided $δ$ is less than an absolute constant. This is sharp wheneve…

Ratio sets of random sets

We study the typical behavior of the size of the ratio set $A/A$ for a random subset $A\subset \{1,\dots , n\}$. For example, we prove that $|A/A|\sim \frac{2\text{Li}_2(3/4)}{π^2}n^2 $ for almost all subsets $A \subset\{1,\dots ,n\}$. We also prove that the proportion of visible lattice points in the lattice $A_1\times\cdots \times A_d$, where $A_i$ is taken at random in $[1,n]$ with $\mathbb P(m\in A_i)=α_i$ for any $m\in [1,n]$, is asymptotic to a constant $μ(α_1,\dots,α_d)$ that involves the polylogarithm of order $d$.

A dichotomy for projections of planar sets

We prove that most one-dimensional projections of a discrete subset of a plane are either dense in R (the real line), or form a discrete subset of R. More precisely, the set E of exceptional directions (for which the indicated dichotomy fails) is a meager subset of the unit circle T of Lebesgue measure 0. The set E however does not need to be small in the sense of Hausdorff dimension.

A note on the combinatorial derivation of non-small sets

Given an infinite group $G$ and a subset $A$ of $G$ we let $Δ(A) = \{g \in G \,:\, |gA \cap A| =\infty\}$ (this is sometimes called the \emph{combinatorial derivation} of $A$). A subset $A$ of $G$ is called: \emph{large} if there exists a finite subset $F$ of $G$ such that $FA=G$; \emph{$Δ$-large} if $Δ(A)$ is large and \emph{small} if for every large subset $L$ of $G$, $(G \setminus A) \cap L$ is large. In this note we show that every non-small set is $Δ$-large, answering a question of Protasov.

Zero-sum subsets of decomposable sets in Abelian groups

A subset $D$ of an Abelian group is $decomposable$ if $\emptyset\ne D\subset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Z\subset D$ with $\sum Z=0$. For every $n\in\mathbb N$ we present a decomposable subset $D$ of cardinality $|D|=n$ in the cyclic group of order $2^n-1$ such that $\sum D=0$, but $\sum T\ne 0$ for any proper non-empty subset $T\subset D$. On the other hand, we prove that every decomposable subset $D\subset\mathbb R$ of cardinality $|D|\le 7$ contains a non-emp…

Polynomial largeness of sumsets and totally ergodic sets

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE set, for any (p(n) \in \Z[n]: °{p(n)} > 0, p(n) \to_{n \to \infty} \infty ) and any subset (B \subset \N ) of positive upper density we have (R_p = A+B \cap \{p(n) | n \in \N \} \neq \emptyset). For (A ) a WM set (subclass of TE sets) we prove that (R_p ) has lower density 1. In addition we obtain a generalization of the latter result to the case of several pol…

Four-vertex traces of finite sets

Let $[n]=X_1\cup X_2\cup X_3$ be a partition with $\lfloor\frac{n}{3}\rfloor \leq |X_i|\leq \lceil\frac{n}{3}\rceil$ and define $\mathcal{G}=\{G\subset [n]\colon |G\cap X_i|\leq 1, 1\leq i\leq 3\}$. It is easy to check that the trace $\mathcal{G}_{\mid Y}:=\{G\cap Y\colon G\in \mathcal{G}\}$ satisfies $|\mathcal{G}_{\mid Y}|\leq 12$ for all 4-sets $Y\subset [n]$. For $n\geq 25$ it is proven that whenever $\mathcal{F}\subset 2^{[n]}$ satisfies $|\mathcal{F}|>|\mathcal{G}|$ then $|\mathcal{F}_{\mid C}|\geq 13$ for some $C\subset [n]$, $|C|=4$. Several further results of a similar flavor are esta…

Theory of dimension for large discrete sets and applications

We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdos and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively.

Four-term progression free sets with three-term progressions in all large subsets

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that all sufficiently large subsets contain at least one such progression. We prove that there exists a positive constant $c$ and a set $A \subset \mathbb F_q^n$ which does not contain a four-term arithmetic progression, with the property that for every subset $A' \subset A$ with $|A'| \geq |A|^{1-c}$, $A'$ contains a nontrivial three term arithmetic progression. We derive this from a more general quantitative Roth-type the…

Additive properties of product sets in an arbitrary finite field

It is proved that for any two subsets $A$ and $B$ of an arbitrary finite field $\Fq$ such that $|A||B|>q$ the identity $16AB=\Fq$ holds. Moreover, it is established that for every subsets $X, Y\subset \Fq$ with the property $|X||Y|\geqslant 2q$ the equality $8XY=\Fq$ holds.

Size of dot product sets determined by pairs of subsets of vector spaces over finite fields

In this paper we study the cardinality of the dot product set generated by two subsets of vector spaces over finite fields. We notice that the results on the dot product problems for one set can be simply extended to two sets. Let E and F be subsets of the d-dimensional vector space F_q^d over a finite field F_q with q elements. As a new result, we prove that if E and F are subsets of the paraboloid and |E||F|\geq C q^d for some large C>1, then |Π(E,F)|\geq c q for some 0<c<1. In particular, we find a connection between the size of the dot product set and the number of lines through both the o…

Difference sets and power residues

Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb{F}_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb{F}_p})^n$ be an arbitrary subset such that $$\{ \mathbf{a}-\mathbf{b}:~\mathbf{a},\mathbf{b}\in A,\mathbf{a}\neq \mathbf{b}\}\cap K^n=\emptyset. $$ Then we prove the exponential upper bound $$ |A|\leq ( p-|K|+ 1 )^n. $$ We use in our proof the linear algebra bound method.

Sum-full sets are not zero-sum-free

Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not zero-sum-free.

不含无大和自由子集的整数集

Sets of integers with no large sum-free subset

该研究解答了Erdos1965年的问题，证明对任意小正数eps，存在n元整数集，其足够大的子集必含满足x+y=z的三元组，丰富了整数集结构相关研究。

Discontinuity and Involutions on Countable Sets

For any infinite subset $X$ of the rationals and a subset $F \subseteq X$ which has no isolated points in $X$ we construct a function $f: X \to X$ such that $f(f(x))=x$ for each $x\in X$ and $F $ is the set of discontinuity points of $f$.

A "hidden" characterization of polyhedral convex sets

We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not = \emptyset$ for any distinct points $x,y\in A$.

Centerpole sets for colorings of Abelian groups

Given a topological group $G$ we calculate or evaluate the cardinal characteristic $c_k(G)$ (and $c_k^B(G)$) equal to the smallest cardinality of a $k$-centerpole subset $C\subset G$ for (Borel) colorings of $G$. A subset $C\subset G$ of a topological group $G$ is called {\em $k$-centerpole} if for each (Borel) $k$-coloring of $G$ there is an unbounded monochromatic subset $G$, which is symmetric with respect to a point $c\in C$ in the sense that $S=cS^{-1}c$.

Sum and Difference Sets in Generalized Dihedral Groups

Given a group $G$, we say that a set $A \subseteq G$ has more sums than differences (MSTD) if $|A+A| > |A-A|$, has more differences than sums (MDTS) if $|A+A| < |A-A|$, or is sum-difference balanced if $|A+A| = |A-A|$. A problem of recent interest has been to understand the frequencies of these type of subsets. The seventh author and Vissuet studied the problem for arbitrary finite groups $G$ and proved that almost all subsets $A\subseteq G$ are sum-difference balanced as $|G|\to\infty$. For the dihedral group $D_{2n}$, they conjectured that of the remaining sets, most are MSTD, i.e., there ar…

Sets Arising as Minimal Additive Complements in the Integers

A subset $C$ of an abelian group $G$ is a minimal additive complement to $W \subseteq G$ if $C + W = G$ and if $C' + W \neq G$ for any proper subset $C' \subset C$. In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that aris…

Direction sets, Lipschitz graphs and density

We consider the direction set determined by various subsets $E$ of Euclidean space and show that there is a trichotomy: Either (i) The subset is the graph of a Lipschitz function and the direction set is not dense in the sphere, (ii) The subset is the graph of a non-Lipschitz function and the direction set is dense but not everything, or (iii) The subset is not a graph (in a suitable sense) and every direction is determined by the set. We then explore a variety of results based on this trichotomy under additional assumptions on the set $E$.

返回集合中的乘积集与对称遍历平均的正性

Product sets in sets of returns and positivity of symmetric ergodic averages

该研究推广了Bergelson等人的结果，证明可和群中足够大的集合包含满足乘积集关系的大子集，还针对有限生成幂零群建立了多项式类似结果。

集合及其子集的拓扑结构

Topological Structures of Sets and their Subsets

本文提出一种基于有限集合及其子集自然拓扑结构的可视化工具，用于构造和研究非普通超图，揭示其拓扑性质、着色、连通性及代数结构。

Maximal discrete sets

We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics $\mathfrak a_ξ$, for $ξ<ω_1$ as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.

On additive properties of sets defined by the Thue-Morse word

In this paper we study some additive properties of subsets of the set $\nats$ of positive integers: A subset $A$ of $\nats$ is called {\it $k$-summable} (where $k\in\ben$) if $A$ contains $\textstyle \big{\sum_{n\in F}x_n | \emp\neq F\subseteq {1,2,...,k\} \big}$ for some $k$-term sequence of natural numbers $x_1<x_2 < ... < x_k$. We say $A \subseteq \nats$ is finite FS-big if $A$ is $k$-summable for each positive integer $k$. We say is $A \subseteq \nats$ is infinite FS-big if for each positive integer $k,$ $A$ contains ${\sum_{n\in F}x_n | \emp\neq F\subseteq \nats and #F\leq k}$ for some in…

离散流形中的离散代数集

Discrete Algebraic sets in Discrete Manifolds

本文证明离散Morse-Sard定理：离散d-流形上任意映射的水平集为(d-k)-流形或空集，并应用于复值函数定义的水平面。

Progression-free sets in Z_4^n are exponentially small

We show that for integer $n>0$, any subset $A \subset Z_4^n$ free of three-term arithmetic progressions has size $|A| < 4^{c n}$, with an absolute constant $c \approx 0.926$.

On product of difference sets for sets of positive density

In this paper we prove that given two sets $E_1,E_2 \subset \mathbb{Z}$ of positive density, there exists $k \geq 1$ which is bounded by a number depending only on the densities of $E_1$ and $E_2$ such that $k\mathbb{Z} \subset (E_1-E_1)\cdot(E_2-E_2)$. As a corollary of the main theorem we deduce that if $α,β> 0$ then there exist $N_0$ and $d_0$ which depend only on $α$ and $β$ such that for every $N \geq N_0$ and $E_1,E_2 \subset \mathbb{Z}_N$ with $|E_1| \geq αN, |E_2| \geq βN$ there exists $d \leq d_0$ a divisor of $N$ satisfying $d \, \mathbb{Z}_N \subset (E_1-E_1)\cdot(E_2-E_2)$.

On the computational complexity of MSTD sets

We outline a general algorithm for verifying whether a subset of the integers is a more sum than differences (MSTD) set, also known as sum dominated sets, and give estimates on its computational complexity. We conclude with some numerical results on large MSTD sets and MSTD subsets of $[1,N]\cap \mathbb Z$ for $N$ up to 160.